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Signals & Systems Lecture 17: Chapter 3 Spectrum Representation
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2 Today's lecture Error signal Convergence of Fourier synthesis Frequency modulation Chirp formula
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3 Convergence of Fourier Synthesis Error Signal: Worst-case error:
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4 Convergence of Fourier Synthesis
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5 General Waveforms Waveforms can be synthesized by the equation x(t) = A 0 + ∑A k cos(2πf k t + k ) These waveforms maybe –constants –cosine signals ( periodic) –complicated-looking signals (not periodic) So far we have dealt with signals whose amplitudes, phases and frequencies do not change with time
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6 Sinusoid with Changing Frequency
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7 Frequency Modulation Most real-world signals exhibit frequency change over time e.g. music. Frequency of a signal may change linearly with time which sounds like a siren or chirp Chirp signal: Signal whose frequency changes linearly with time from some low value to high value Let x(t) =Acos(ω 0 t + ) =Acos ψ(t) ψ(t) = ω 0 t + and dψ(t)/dt = ω 0 where ψ(t) denotes the time varying angle function
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8 Frequency Modulation We can create a signal with quadratic angle function by defining ψ(t) = 2πμt 2 + 2πf 0 t + instantaneous frequency = slope of the angle function ω i = dψ(t)/dt f i (t) = 1/2 π dψ(t)/dt f i (t) = 2μt + f 0
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9 Example 3.8: Synthesize a Chirp Formula Synthesize a frequency sweep from f 1 = 220 Hz to f 2 = 2320 Hz over a 3-second time interval. f i (t) = (f 2 - f 1 )t / T 2 + f 1 ψ i (t) = ∫ ω i (u) du 0 t
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10 Frequency Modulation: Chirp Signals
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